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Theorem ltbtwnnq 8602
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltbtwnnq  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltbtwnnq
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 8550 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4737 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simprd 449 . . 3  |-  ( A 
<Q  B  ->  B  e. 
Q. )
4 ltexnq 8599 . . . 4  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. y
( A  +Q  y
)  =  B ) )
5 eleq1 2343 . . . . . . . . . 10  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  y
)  e.  Q.  <->  B  e.  Q. ) )
65biimparc 473 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  +Q  y )  e.  Q. )
7 addnqf 8572 . . . . . . . . . . 11  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5394 . . . . . . . . . 10  |-  dom  +Q  =  ( Q.  X.  Q. )
9 0nnq 8548 . . . . . . . . . 10  |-  -.  (/)  e.  Q.
108, 9ndmovrcl 6006 . . . . . . . . 9  |-  ( ( A  +Q  y )  e.  Q.  ->  ( A  e.  Q.  /\  y  e.  Q. ) )
116, 10syl 15 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  e. 
Q.  /\  y  e.  Q. ) )
1211simprd 449 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  y  e.  Q. )
13 nsmallnq 8601 . . . . . . . 8  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
1411simpld 445 . . . . . . . . . . . 12  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  A  e.  Q. )
151brel 4737 . . . . . . . . . . . . 13  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
1615simpld 445 . . . . . . . . . . . 12  |-  ( z 
<Q  y  ->  z  e. 
Q. )
17 ltaddnq 8598 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1814, 16, 17syl2an 463 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  A  <Q  ( A  +Q  z ) )
19 ltanq 8595 . . . . . . . . . . . . . 14  |-  ( A  e.  Q.  ->  (
z  <Q  y  <->  ( A  +Q  z )  <Q  ( A  +Q  y ) ) )
2019biimpa 470 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  <Q  ( A  +Q  y ) )
2114, 20sylan 457 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
22 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  y )  =  B )
2321, 22breqtrd 4047 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  B )
24 ovex 5883 . . . . . . . . . . . 12  |-  ( A  +Q  z )  e. 
_V
25 breq2 4027 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
26 breq1 4026 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2725, 26anbi12d 691 . . . . . . . . . . . 12  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2824, 27spcev 2875 . . . . . . . . . . 11  |-  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
2918, 23, 28syl2anc 642 . . . . . . . . . 10  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
3029ex 423 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3130exlimdv 1664 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( E. z 
z  <Q  y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3213, 31syl5 28 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
3312, 32mpd 14 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
3433ex 423 . . . . 5  |-  ( B  e.  Q.  ->  (
( A  +Q  y
)  =  B  ->  E. x ( A  <Q  x  /\  x  <Q  B ) ) )
3534exlimdv 1664 . . . 4  |-  ( B  e.  Q.  ->  ( E. y ( A  +Q  y )  =  B  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
364, 35sylbid 206 . . 3  |-  ( B  e.  Q.  ->  ( A  <Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
373, 36mpcom 32 . 2  |-  ( A 
<Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
38 ltsonq 8593 . . . 4  |-  <Q  Or  Q.
3938, 1sotri 5070 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4039exlimiv 1666 . 2  |-  ( E. x ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4137, 40impbii 180 1  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   Q.cnq 8474    +Q cplq 8477    <Q cltq 8480
This theorem is referenced by:  nqpr  8638  reclem2pr  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542
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